Theorem 1 (Upper bound of \(k\)  for monotone \(v(S)\)  [1]}) 

Suppose that there exists some constant \(\eta \)  s.t. \(k(x,x^{\prime })\le \eta \le k^*\)  for all \(x,x^{\prime }\in T\)  and \(x\ne x^{\prime }\) . Then,
\(\forall S\subseteq T\ \ [\eta = {tk^*}/({(s+1)(s(t-2)+t)})]\Rightarrow \\ [\forall x\in T\setminus S\ \ v(S\cup \lbrace x\rbrace )\ge v(S)]\ .\) 

